The proof is similar to Lemma 3 of [1]}. For completeness, we repeat it again.
Let \(f^\star \)  denotes the density function maximizing \(\mathbb {E}[e^{\lambda m \Delta ^2} ]\)  subject to the constraint that \(\mathbb {P}[\Delta \ge \epsilon ]\le \exp (\frac{-m\epsilon ^2}{2\sigma ^2})\) . The maximum occurs when \(\mathbb {P}_{f^\star }[\Delta \ge \epsilon ]= \exp (\frac{-m\epsilon ^2}{2\sigma ^2})\)  leading to
\(f^\star (\Delta )=\frac{m\Delta }{\sigma ^2} \exp \left( -\frac{m\Delta ^2}{2\sigma ^2} \right)\mathbb {1} \lbrace  \Delta \ge 0 \rbrace .\) 
